Abstract

We describe a novel method for the measurement of spatially two-dimensional temporal amplitude correlation function of two coherent short light pulses with picosecond resolution in which the light pulses are recorded in a volume-holographic medium and the recorded information is read out by direct measurement of the spatial distribution of the grating formed in the medium. The feasibility of this method was experimentally proved by means of a photorefractive LiNbO3:Fe crystal and a 3.5-ps frequency-doubled mode-locked Nd:YAG laser. We measured the diffraction of a cw He–Ne laser by the grating formed in the crystal and calculated its envelope function by inversely solving coupled-wave equations.

© 1997 Optical Society of America

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  1. N. Abramson, “Light-in-flight recording by holography,” Opt. Lett. 3, 121 (1978).
    [CrossRef] [PubMed]
  2. A. A. Bugaev and B. P. Zakharchenya, “Holographic time diagnostics with picosecond resolution,” Opt. Spectrosc. 60, 646 (1986).
  3. Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101 (1990).
    [CrossRef]
  4. C. Joubert, M. L. Roblin, and R. Grousson, “Temporal reversal of picosecond optical pulses by holographic phase conjugation,” Appl. Opt. 28, 4604 (1989).
    [CrossRef] [PubMed]
  5. L. H. Acioli, M. Ulman, E. P. Ippen, J. G. Fujimoto, H. Kong, B. S. Chen, and M. Cronin-Golomb, “Femtosecond temporal encoding in barium titanate,” Opt. Lett. 16, 1984 (1991).
    [CrossRef] [PubMed]
  6. K. Ema, M. Kuwata-Gonokami, and F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nolinearity,” Appl. Phys. Lett. 59, 2799 (1991).
    [CrossRef]
  7. R. Trebino, C. Hayden, A. M. Johnson, W. M. Simpson, and A. M. Levine, “Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses,” Opt. Lett. 15, 1079 (1990).
    [CrossRef] [PubMed]
  8. A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.
  9. V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521 (1990).
    [CrossRef]
  10. X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7, 2347 (1990).
    [CrossRef]
  11. G. C. Valley, “Short-pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637 (1983).
    [CrossRef]
  12. A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, and T. F. Boggess, “Formation, decay, and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606 (1989).
    [CrossRef]
  13. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
    [CrossRef]
  14. T. F. Boggess, J. O. White, and G. C. Valley, “Two-photon absorption and anisotropic transient energy transfer in BaTiO3 with 1-ps excitation,” J. Opt. Soc. Am. B 7, 2255 (1990).
    [CrossRef]
  15. P. Günter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), p. 53.
  16. M. B. Klein in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), Chap. 7, p. 220.
  17. J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
    [CrossRef]
  18. G. Pauliat and G. Roosen, “Photorefractive effect generated in sillenite crystals by picosecond pulses and comparison with the quasi-continuous regime,” J. Opt. Soc. Am. B 7, 2259 (1990).
    [CrossRef]
  19. K. Kurtz and D. von der Linde, “Nonlinear optical excitation of photovoltaic LiNbO3,” Ferroelectrics 21, 621 (1978).
    [CrossRef]
  20. S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric and optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797 (1968).
    [CrossRef]
  21. D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
    [CrossRef]
  22. H. Okamura and K. Kuroda, “A method for the evaluation of the grating envelope in volume holographic media,” Appl. Phys. B 62, 399 (1996).
    [CrossRef]
  23. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).
    [CrossRef]

1996 (1)

H. Okamura and K. Kuroda, “A method for the evaluation of the grating envelope in volume holographic media,” Appl. Phys. B 62, 399 (1996).
[CrossRef]

1991 (2)

L. H. Acioli, M. Ulman, E. P. Ippen, J. G. Fujimoto, H. Kong, B. S. Chen, and M. Cronin-Golomb, “Femtosecond temporal encoding in barium titanate,” Opt. Lett. 16, 1984 (1991).
[CrossRef] [PubMed]

K. Ema, M. Kuwata-Gonokami, and F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nolinearity,” Appl. Phys. Lett. 59, 2799 (1991).
[CrossRef]

1990 (6)

1989 (2)

1986 (2)

A. A. Bugaev and B. P. Zakharchenya, “Holographic time diagnostics with picosecond resolution,” Opt. Spectrosc. 60, 646 (1986).

J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
[CrossRef]

1983 (1)

G. C. Valley, “Short-pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637 (1983).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

1978 (2)

K. Kurtz and D. von der Linde, “Nonlinear optical excitation of photovoltaic LiNbO3,” Ferroelectrics 21, 621 (1978).
[CrossRef]

N. Abramson, “Light-in-flight recording by holography,” Opt. Lett. 3, 121 (1978).
[CrossRef] [PubMed]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).
[CrossRef]

1974 (1)

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

1968 (1)

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric and optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797 (1968).
[CrossRef]

Abramson, N.

Acioli, L. H.

Boggess, T. F.

Bohnert, K.

Bugaev, A. A.

A. A. Bugaev and B. P. Zakharchenya, “Holographic time diagnostics with picosecond resolution,” Opt. Spectrosc. 60, 646 (1986).

Bylsma, R. B.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.

Camlibel, I.

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric and optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797 (1968).
[CrossRef]

Chen, B. S.

Cronin-Golomb, M.

Didomenico, D.

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric and optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797 (1968).
[CrossRef]

Dominic, V.

V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521 (1990).
[CrossRef]

X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7, 2347 (1990).
[CrossRef]

Ema, K.

K. Ema, M. Kuwata-Gonokami, and F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nolinearity,” Appl. Phys. Lett. 59, 2799 (1991).
[CrossRef]

Feinberg, J.

V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521 (1990).
[CrossRef]

X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7, 2347 (1990).
[CrossRef]

Ferrier, J. L.

J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
[CrossRef]

Fujimoto, J. G.

Gazengel, J.

J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
[CrossRef]

Glass, A. M.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.

Grousson, R.

Günter, P.

P. Günter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), p. 53.

Hayden, C.

Huignard, J.-P.

P. Günter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), p. 53.

Ippen, E. P.

Johnson, A. M.

R. Trebino, C. Hayden, A. M. Johnson, W. M. Simpson, and A. M. Levine, “Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses,” Opt. Lett. 15, 1079 (1990).
[CrossRef] [PubMed]

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.

Joubert, C.

Klein, M. B.

M. B. Klein in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), Chap. 7, p. 220.

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).
[CrossRef]

Kong, H.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

Kuroda, K.

H. Okamura and K. Kuroda, “A method for the evaluation of the grating envelope in volume holographic media,” Appl. Phys. B 62, 399 (1996).
[CrossRef]

Kurtz, K.

K. Kurtz and D. von der Linde, “Nonlinear optical excitation of photovoltaic LiNbO3,” Ferroelectrics 21, 621 (1978).
[CrossRef]

Kuwata-Gonokami, M.

K. Ema, M. Kuwata-Gonokami, and F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nolinearity,” Appl. Phys. Lett. 59, 2799 (1991).
[CrossRef]

Levine, A. M.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

Mazurenko, Y. T.

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101 (1990).
[CrossRef]

Mikulyak, R. M.

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

Mullen, R. A.

Nelson, D. F.

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

Nguyen Phu, X.

J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

Okamura, H.

H. Okamura and K. Kuroda, “A method for the evaluation of the grating envelope in volume holographic media,” Appl. Phys. B 62, 399 (1996).
[CrossRef]

Olson, D. H.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.

Pauliat, G.

Pierce, R. M.

V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521 (1990).
[CrossRef]

Rivoire, G.

J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
[CrossRef]

Roblin, M. L.

Roosen, G.

Shimizu, F.

K. Ema, M. Kuwata-Gonokami, and F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nolinearity,” Appl. Phys. Lett. 59, 2799 (1991).
[CrossRef]

Simpson, W. M.

R. Trebino, C. Hayden, A. M. Johnson, W. M. Simpson, and A. M. Levine, “Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses,” Opt. Lett. 15, 1079 (1990).
[CrossRef] [PubMed]

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.

Smirl, A. L.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

Trebino, R.

Ulman, M.

Valley, G. C.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

von der Linde, D.

K. Kurtz and D. von der Linde, “Nonlinear optical excitation of photovoltaic LiNbO3,” Ferroelectrics 21, 621 (1978).
[CrossRef]

Wemple, S. H.

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric and optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797 (1968).
[CrossRef]

White, J. O.

Yao, X. S.

X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7, 2347 (1990).
[CrossRef]

V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521 (1990).
[CrossRef]

Zakharchenya, B. P.

A. A. Bugaev and B. P. Zakharchenya, “Holographic time diagnostics with picosecond resolution,” Opt. Spectrosc. 60, 646 (1986).

Appl. Opt. (1)

Appl. Phys. B (2)

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101 (1990).
[CrossRef]

H. Okamura and K. Kuroda, “A method for the evaluation of the grating envelope in volume holographic media,” Appl. Phys. B 62, 399 (1996).
[CrossRef]

Appl. Phys. Lett. (2)

K. Ema, M. Kuwata-Gonokami, and F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nolinearity,” Appl. Phys. Lett. 59, 2799 (1991).
[CrossRef]

V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521 (1990).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).
[CrossRef]

Ferroelectrics (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979); “Holographic storage in electrooptic crystals. II. Beam coupling-light amplification,”  22, 961 (1979).
[CrossRef]

K. Kurtz and D. von der Linde, “Nonlinear optical excitation of photovoltaic LiNbO3,” Ferroelectrics 21, 621 (1978).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. C. Valley, “Short-pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637 (1983).
[CrossRef]

J. Appl. Phys. (1)

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. Chem. Solids (1)

S. H. Wemple, D. Didomenico, and I. Camlibel, “Dielectric and optical properties of melt-grown BaTiO3,” J. Phys. Chem. Solids 29, 1797 (1968).
[CrossRef]

Opt. Commun. (1)

J. L. Ferrier, J. Gazengel, X. Nguyen Phu, and G. Rivoire, “Picosecond holography and four-wave mixing in BSO,” Opt. Commun. 58, 343 (1986).
[CrossRef]

Opt. Lett. (3)

Opt. Spectrosc. (1)

A. A. Bugaev and B. P. Zakharchenya, “Holographic time diagnostics with picosecond resolution,” Opt. Spectrosc. 60, 646 (1986).

Other (3)

P. Günter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), p. 53.

M. B. Klein in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer, Berlin, 1988), Chap. 7, p. 220.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting Vol. II of 1988 OSA Technical Digest (Optical Society of America, Washington, D.C., 1988), paper THC4, p. 128.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the principle. Object and sampling pulses counterpropagate in a thick recording medium. If the width of the sampling pulse is short compared with that of the object pulse, the envelope of the object pulse is mapped in the medium as an envelope function of the grating.

Fig. 2
Fig. 2

Schematic of the amplitude correlation of the intersecting pulses. The grating having wave vector K is formed through interference between pulses 1 and 2, and the modulation of the grating at position P is proportional to the cross-correlation function between the hatched areas in the pulses. The portion that is cross correlated and the delay for the correlation are determined uniquely, depending on the position P.

Fig. 3
Fig. 3

Result of the computation for two-beam coupling of counterpropagating pulses. The ordinates are amplitude-correlation-function normalized by the averaged total intensity and are proportional to the magnitude of the grating at steady state. The contributions from (a) the original pulse and (b) the perturbation through two-beam coupling are shown. The coupling coefficient is 10 cm-1, and the group refractive index is 2.2. The computation was carried out for pulse durations of 1, 2, and 4 ps.

Fig. 4
Fig. 4

Experimental setup for recording and readout of short pulses. 3.5-ps pulses from a frequency-doubled mode-locked Nd:YAG laser (λ=532 nm) form a grating in the LiNbO3:Fe crystal. This grating is read out by a cw He–Ne laser beam (λ=633 nm), and the amplitude and the relative phase of the diffracted wave are measured by heterodyne detection by means of two acousto-optic modulators (AOM’s) as a function of the deviation from the Bragg angle. The crystal was scanned horizontally and vertically to achieve three-dimensional measurement of the grating envelope.

Fig. 5
Fig. 5

(a) Amplitude and (b) relative phase of the measured correlation function. In both plots the horizontal axis is the position in the cross section of the light pulse, and the vertical axis is the time delay of the correlation. In (a) the dark areas represent large magnitude, and the dashed lines denote the boundaries of the measurable area.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ESC0τpI1(t)dt,
ESC0τpI1(t)dt0τpI0(t)dt.
ΔnESC0τpA1(r, t)A2*(r, t)dt,
F(r)0TA1(r, t)A2*(r, t)dt=exp(iK·r)0Tf1(r-v1t)×f2*(r-v2t)dt,
F(r)=exp(iK·r)0Tf1[r-v1(τ-t)]U*(t)dt,
SL(δ)=(iβ)/40lg(r)R(r; δ)exp(irδ)dr,
k1k·+n1ctA1(r, t)=η12G(r)I0(r)A2(r, t),
k2k·+n2ctA2(r, t)=-η21*G*(r)I0(r)A1(r, t),
G(r)=A1(r, t)A2*(r, t)=1T-T/2T/2A1(r, t)A2*(r, t)dt,
z+1vtA1(z, t)=ηG(z)I0A2(z, t),
-z+1vtA2(z, t)=-η*G*(z)I0A1(z, t),
1vtA1(ζ+, t)=ηG(ζ++vt)I0A2(ζ++2vt, t),
1vtA2(ζ-, t)=-η*G*(ζ--vt)I0A1(ζ--2vt, t).
Ai(ζ, t)=Ai0(ζ)+Ai1(ζ, t)(i=1, 2),
G(z)=G0(z)+G1(z),
G0(z)=A10A20*,
A11(ζ+, t)=vηI0-T/2tG0(ζ++vt)A20(ζ++2vt)dt,
A21(ζ-, t)=-vη*I0-T/2tG0*(ζ--vt)×A10(ζ--2vt)dt,
G1(z)=A10A21*+A11A20*=vηI0T--T/2T/2dtA10(z-vt)×-T/2τG0(z+vt-vt)×A10*(z+vt-2vt)dt+-T/2T/2dtA20*(z+vt)-T/2τG0(z-vt+vt)×A20(z-vt+2vt)dt.

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